reserve x, y, z for object,
  i, j, n for Nat,
  p, q, r for FinSequence,
  v for FinSequence of NAT;

theorem Th1:
  for i,j being Nat
  st elementary_tree i c= elementary_tree j holds i <= j
proof
  let i,j be Nat;
  assume that
A1: elementary_tree i c= elementary_tree j and
A2: i > j;
 reconsider j as Element of NAT by ORDINAL1:def 12;
 <*j*> in elementary_tree i by A2,TREES_1:28;
then A3: ex n being Nat st n < j & <*j*> = <*n*> by A1,TREES_1:30;
 <*j*>.1 = j;
  hence thesis by A3,FINSEQ_1:40;
end;
